Reduction of topological stable rank in inductive limits of C^∗-algebras

Marius Dadarlat, Gabriel Nagy, András Némethi and Cornel Pasnicu

Vol. 153 (1992), No. 2, 267–276

DOI: 10.2140/pjm.1992.153.267

Abstract

We consider inductive limits A of sequences A₁ → A₂ →⋯ finite direct sums of C^∗-algebras of continuous functions from compact Hausdorff spaces into full matrix algebras. We prove that A has topological stable rank (tsr) one provided that A is simple and the sequence of the dimensions of the spectra of A_i is bounded. For unital A, tsr(A) = 1 means that the set of invertible elements is dense in A. If A is infinite dimensional, then the simplicity of A implies that the sizes of the involved matrices tend to infinity, so by general arguments one gets tsr(A_i) ≤ 2 for large enough i whence tsr(A) ≤ 2. The reduction of tsr from two to one requires arguments which are strongly related to this special class of C^∗-algebras.

Mathematical Subject Classification 2000

Primary: 46L80

Secondary: 19B10, 19K99

Milestones

Received: 5 October 1990

Revised: 7 June 1991

Published: 1 April 1992